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Properties

Label 12-3e36-1.1-c1e6-0-9
Degree $12$
Conductor $1.501\times 10^{17}$
Sign $1$
Analytic cond. $38907.0$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 9·4-s − 3·5-s + 18·8-s − 9·10-s + 12·11-s + 36·16-s + 9·17-s + 3·19-s − 27·20-s + 36·22-s − 3·23-s + 9·25-s + 12·29-s + 66·32-s + 27·34-s + 3·37-s + 9·38-s − 54·40-s + 24·41-s + 108·44-s − 9·46-s + 30·47-s + 27·50-s − 36·53-s − 36·55-s + 36·58-s + ⋯
L(s)  = 1  + 2.12·2-s + 9/2·4-s − 1.34·5-s + 6.36·8-s − 2.84·10-s + 3.61·11-s + 9·16-s + 2.18·17-s + 0.688·19-s − 6.03·20-s + 7.67·22-s − 0.625·23-s + 9/5·25-s + 2.22·29-s + 11.6·32-s + 4.63·34-s + 0.493·37-s + 1.45·38-s − 8.53·40-s + 3.74·41-s + 16.2·44-s − 1.32·46-s + 4.37·47-s + 3.81·50-s − 4.94·53-s − 4.85·55-s + 4.72·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{36}\)
Sign: $1$
Analytic conductor: \(38907.0\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{36} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(52.75729872\)
\(L(\frac12)\) \(\approx\) \(52.75729872\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 3 T + 9 T^{3} - 9 T^{4} - 3 p^{2} T^{5} + 37 T^{6} - 3 p^{3} T^{7} - 9 p^{2} T^{8} + 9 p^{3} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 3 T - 9 T^{4} - 3 p T^{5} + 109 T^{6} - 3 p^{2} T^{7} - 9 p^{2} T^{8} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 11 T^{3} - 27 T^{4} - 27 T^{5} + 345 T^{6} - 27 p T^{7} - 27 p^{2} T^{8} + 11 p^{3} T^{9} + p^{6} T^{12} \)
11 \( 1 - 12 T + 54 T^{2} - 9 p T^{3} + 207 T^{4} - 2361 T^{5} + 12241 T^{6} - 2361 p T^{7} + 207 p^{2} T^{8} - 9 p^{4} T^{9} + 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 29 T^{3} - 81 T^{4} - 621 T^{5} + 1641 T^{6} - 621 p T^{7} - 81 p^{2} T^{8} + 29 p^{3} T^{9} + p^{6} T^{12} \)
17 \( ( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{3} \)
19 \( 1 - 3 T - 24 T^{2} + 131 T^{3} + 117 T^{4} - 1116 T^{5} + 3003 T^{6} - 1116 p T^{7} + 117 p^{2} T^{8} + 131 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 3 T + 54 T^{3} - 441 T^{4} - 2481 T^{5} + 3889 T^{6} - 2481 p T^{7} - 441 p^{2} T^{8} + 54 p^{3} T^{9} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 12 T + 81 T^{2} - 243 T^{3} + 1125 T^{4} - 8445 T^{5} + 70210 T^{6} - 8445 p T^{7} + 1125 p^{2} T^{8} - 243 p^{3} T^{9} + 81 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 27 T^{2} + 245 T^{3} - 945 T^{4} - 5967 T^{5} + 71382 T^{6} - 5967 p T^{7} - 945 p^{2} T^{8} + 245 p^{3} T^{9} - 27 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 3 T - 78 T^{2} + 5 p T^{3} + 3681 T^{4} - 4194 T^{5} - 141339 T^{6} - 4194 p T^{7} + 3681 p^{2} T^{8} + 5 p^{4} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 24 T + 270 T^{2} - 2043 T^{3} + 13545 T^{4} - 87621 T^{5} + 560953 T^{6} - 87621 p T^{7} + 13545 p^{2} T^{8} - 2043 p^{3} T^{9} + 270 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 27 T^{2} + 389 T^{3} - 1701 T^{4} - 12987 T^{5} + 148926 T^{6} - 12987 p T^{7} - 1701 p^{2} T^{8} + 389 p^{3} T^{9} - 27 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 30 T + 405 T^{2} - 3249 T^{3} + 14895 T^{4} - 9255 T^{5} - 267722 T^{6} - 9255 p T^{7} + 14895 p^{2} T^{8} - 3249 p^{3} T^{9} + 405 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 18 T + 240 T^{2} + 1989 T^{3} + 240 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 3 T + 27 T^{2} - 783 T^{3} + 612 T^{4} + 714 T^{5} + 588493 T^{6} + 714 p T^{7} + 612 p^{2} T^{8} - 783 p^{3} T^{9} + 27 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 9 T + 144 T^{2} - 988 T^{3} + 15039 T^{4} - 96363 T^{5} + 1055181 T^{6} - 96363 p T^{7} + 15039 p^{2} T^{8} - 988 p^{3} T^{9} + 144 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 18 T + 117 T^{2} + 29 T^{3} - 189 T^{4} + 85995 T^{5} + 1190478 T^{6} + 85995 p T^{7} - 189 p^{2} T^{8} + 29 p^{3} T^{9} + 117 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 9 T + 30 T^{2} - 99 T^{3} - 3531 T^{4} - 5580 T^{5} + 200671 T^{6} - 5580 p T^{7} - 3531 p^{2} T^{8} - 99 p^{3} T^{9} + 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 6 T - 114 T^{2} - 58 T^{3} + 8676 T^{4} - 27576 T^{5} - 826869 T^{6} - 27576 p T^{7} + 8676 p^{2} T^{8} - 58 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 27 T + 270 T^{2} + 686 T^{3} - 8775 T^{4} - 98577 T^{5} - 753063 T^{6} - 98577 p T^{7} - 8775 p^{2} T^{8} + 686 p^{3} T^{9} + 270 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 6 T + 81 T^{2} + 99 T^{3} + 6147 T^{4} - 53139 T^{5} - 61226 T^{6} - 53139 p T^{7} + 6147 p^{2} T^{8} + 99 p^{3} T^{9} + 81 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 78 T^{2} - 1998 T^{3} - 858 T^{4} + 77922 T^{5} + 1866463 T^{6} + 77922 p T^{7} - 858 p^{2} T^{8} - 1998 p^{3} T^{9} - 78 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 + 324 T^{2} + 686 T^{3} + 55188 T^{4} + 166860 T^{5} + 6246579 T^{6} + 166860 p T^{7} + 55188 p^{2} T^{8} + 686 p^{3} T^{9} + 324 p^{4} T^{10} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.69609799201081114586181535754, −5.55472309760831013214270384438, −5.26912491901641081810838465170, −4.97531739829645053589892108330, −4.61309373298222874294605468836, −4.59950112280656088816638426058, −4.36025548447866173911538772970, −4.28469084225015372729719388170, −4.24958985855286868250437333755, −4.17183283170238557999232869913, −3.72944367143785401910024213266, −3.70073926471724745582060194903, −3.38198641614930659633633537006, −3.26996753321113511141156833973, −2.97578365041511978315825383235, −2.91645024259358726082734741659, −2.88251453772076108895226351368, −2.57161396742161245230964763244, −2.40615078735739029379176355235, −1.97915308860002905321069485065, −1.56299403447934425558187594264, −1.39664286449151233890591032924, −1.24079584972774686738362277000, −0.930330046729912974723155318378, −0.896785076849626519119223588539, 0.896785076849626519119223588539, 0.930330046729912974723155318378, 1.24079584972774686738362277000, 1.39664286449151233890591032924, 1.56299403447934425558187594264, 1.97915308860002905321069485065, 2.40615078735739029379176355235, 2.57161396742161245230964763244, 2.88251453772076108895226351368, 2.91645024259358726082734741659, 2.97578365041511978315825383235, 3.26996753321113511141156833973, 3.38198641614930659633633537006, 3.70073926471724745582060194903, 3.72944367143785401910024213266, 4.17183283170238557999232869913, 4.24958985855286868250437333755, 4.28469084225015372729719388170, 4.36025548447866173911538772970, 4.59950112280656088816638426058, 4.61309373298222874294605468836, 4.97531739829645053589892108330, 5.26912491901641081810838465170, 5.55472309760831013214270384438, 5.69609799201081114586181535754

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.